Given, a = 2b

Thus,  === 3

If then

For real root,

Divisibility of 2 : Unit digit of a number is 0,2,4,6 and 8 then the number is divisible by 2.

b = 0 not possible because number is not divisible by 5

Divisibility of 11: Difference between sum of digits at even places and sum of digits at odd places is either 0 or multiple of 11.

If b =2 then,

(2+4+5) – (3+a+2) = 0 or multiple of 11

6 – a =0

a = 6

Number = 234652

If b = 8 then,

| (2+4+5) – (3+a+8) |= 0 or multiple of 11

| 11 – 11 – a |=0

a =0

Number = 234058

Greatest number = 234652

According to the question,

 400 ×

 400 ×

 100 – x =

 100 – x = 60

 x = 100 – 60

 x = 40

Therefore, the value of x = 40

Hence, option B is correct.

We know that:

Mean =

⇒ 21 =

⇒ 84 = 7x + 21

⇒ 7x = 84 – 21

⇒ x =

⇒ x = 9

Hence, option D is correct.

(77⁷⁷ + 77) ÷ 78

⇒ [(77⁷⁷ + 1) + 76]/78

⇒ [(77⁷⁷ + 1)/78 + 76/78

⇒ [(77⁷⁷ + 1⁷⁷)/78 + 76/78

We know, (xⁿ + yⁿ) is divisible by (x + y)
∴ (77⁷⁷ + 1⁷⁷) is divisible by 78.

∴ Remainder = 76

In ΔABC, points D and E are on AB and AC, respectively, such that DE is parallel to BC. If AD = 3 cm, BD=6 cm and AE = 2 cm as shown in figure below:

Let CE = x cm

If DE is parallel to BC

Then

⇒

Hence, CE = 4 cm

In Δ ABD and Δ CAD

∠ ADB = ∠ ADC = 90° each

∠ BAD = ∠ ACD

= 90° – ∠ B and AD = AD

∴ ADB ∼ Δ CAD and

In Δ ADB = ∠ CAB

∠ ADB = ∠ BAC = 90° each

And ∠ ABC = ∠ ABD

∴ Δ ADB ∼ Δ CAB

Here, (i) and (ii) are correct statements.

Inner circumference of circular path – 264 m

Inner radius =

Outer radius = 42m + 3m = 45 m

Area of path =

=

Let be an event. Then, 

Therefore,  cannot be more than 1.

Option D is the correct answer.

=

=

=

=

The difference between the answers = =

According to the given condition

0° < A  90°

cos A = 60°